# Representation and Analysis of Piecewise Linear Functions in Abs-Normal Form

Abstract : It follows from the well known min/max representation given by Scholtes in his recent Springer book, that all piecewise linear continuous functions $y = F(x) : \mathbb {R}^n \rightarrow \mathbb {R}^m$ can be written in a so-called abs-normal form. This means in particular, that all nonsmoothness is encapsulated in $s$ absolute value functions that are applied to intermediate switching variables $z_i$ for $i=1, \ldots ,s$ . The relation between the vectors $x, z$ , and $y$ is described by four matrices $Y, L, J$ , and $Z$ , such that $\left[ \begin{array}{c} z \\ y \end{array}\right] = \left[ \begin{array}{c} c \\ b \end{array}\right] + \left[ \begin{array}{cc} Z &{} L \\ J &{} Y \end{array}\right] \left[ \begin{array}{c} x \\ |z |\end{array}\right]$ This form can be generated by ADOL-C or other automatic differentation tools. Here $L$ is a strictly lower triangular matrix, and therefore $z_i$ can be computed successively from previous results. We show that in the square case $n=m$ the system of equations $F(x) = 0$ can be rewritten in terms of the variable vector $z$ as a linear complementarity problem (LCP). The transformation itself and the properties of the LCP depend on the Schur complement $S = L - Z J^{-1} Y$ .
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Christian Pötzsche; Clemens Heuberger; Barbara Kaltenbacher; Franz Rendl. 26th Conference on System Modeling and Optimization (CSMO), Sep 2013, Klagenfurt, Austria. Springer Berlin Heidelberg, IFIP Advances in Information and Communication Technology, AICT-443, pp.327-336, 2014, System Modeling and Optimization. 〈10.1007/978-3-662-45504-3_32〉
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Tom Streubel, Andreas Griewank, Manuel Radons, Jens-Uwe Bernt. Representation and Analysis of Piecewise Linear Functions in Abs-Normal Form. Christian Pötzsche; Clemens Heuberger; Barbara Kaltenbacher; Franz Rendl. 26th Conference on System Modeling and Optimization (CSMO), Sep 2013, Klagenfurt, Austria. Springer Berlin Heidelberg, IFIP Advances in Information and Communication Technology, AICT-443, pp.327-336, 2014, System Modeling and Optimization. 〈10.1007/978-3-662-45504-3_32〉. 〈hal-01286443〉

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